10.1 The binary’s multipole moments

The general expressions of the source multipole moments given by Theorem 6, Equations (85), are first
to be worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the
formula (91), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in
Equations (115) by means of the retarded-type elementary potentials (117, 118, 119). Then we specialize
each of the (quite numerous) terms to the case of point-particle binaries by inserting, for the matter
stress-energy tensor , the standard expression made out of Dirac delta-functions. The infinite self-field
of point-particles is removed by means of the Hadamard regularization; and dimensional regularization is
used to compute the few ambiguity parameters (see Section 8). This computation has been performed
in [49] at the 1PN order, and in [33] at the 2PN order; we report below the most accurate 3PN results
obtained in Refs. [45, 44, 31, 32].

The difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the
particle’s positions and velocities, of many non-linear integrals. We refer to [45] for full details; nevertheless,
let us give a few examples of the type of technical formulas that are employed in this calculation. Typically
we have to compute some integrals like

where and . When and , this integral is perfectly
well-defined (recall that the finite part deals with the bound at infinity). When or
, our basic ansatz is that we apply the definition of the Hadamard partie finie provided by
Equation (124). Two examples of closed-form formulas that we get, which do not necessitate the Hadamard
partie finie, are (quadrupole case )

We denote for example (and ); the constant is the one pertaining to
the finite-part process (see Equation (36)). One example where the integral diverges at the location of the
particle 1 is

where is the Hadamard-regularization constant introduced in
Equation (124)37.

The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment ,
since this moment necessitates the full 3PN precision. The result of Ref. [45] for this moment (in the case of
circular orbits) is

where we pose and . The third term is the 2.5PN radiation-reaction term,
which does not contribute to the energy flux for circular orbits. The two important coefficients are and
, whose expressions through 3PN order are

These expressions are valid in harmonic coordinates via the post-Newtonian parameter given by
Equation (188). As we see, there are two types of logarithms in the moment: One type involves the length
scale related by Equation (184) to the two gauge constants and present in the 3PN equations
of motion; the other type contains the different length scale coming from the general formalism of
Part A - indeed, recall that there is a operator in front of the source multipole moments in
Theorem 6. As we know, that is pure gauge; it will disappear from our physical results at the
end. On the other hand, we have remarked that the multipole expansion outside a general
post-Newtonian source is actually free of , since the ’s present in the two terms of
Equation (67) cancel out. We shall indeed find that the constants present in Equations (220) are
compensated by similar constants coming from the non-linear wave “tails of tails”. Finally, the
constants , , and are the Hadamard-regularization ambiguity parameters which take the
values (136).

Besides the 3PN mass quadrupole (219, 220), we need also the mass octupole moment
and current quadrupole moment , both of them at the 2PN order; these are given by [45]

Also needed are the 1PN mass -pole, 1PN current -pole (octupole), Newtonian mass -pole and
Newtonian current -pole:

These results permit one to control what can be called the “instantaneous” part, say , of the total
energy flux, by which we mean that part of the flux that is generated solely by the source multipole
moments, i.e. not counting the “non-instantaneous” tail integrals. The instantaneous flux is defined by the
replacement into the general expression of given by Equation (60) of all the radiative moments
and by the corresponding (th time derivatives of the) source moments and . Actually, we
prefer to define by means of the intermediate moments and . Up to the 3.5PN order we
have

The time derivatives of the source moments (219, 220, 221, 222) are computed by means of the
circular-orbit equations of motion given by Equation (189, 190); then we substitute them into
Equation (223)38.
The net result is

The Newtonian approximation, , is the prediction of the Einstein quadrupole
formula (4), as computed by Landau and Lifchitz [153]. In Equation (224), we have replaced the
Hadamard regularization ambiguity parameters and arising at the 3PN order by their values (135)
and (137).